Thursday, September 14, 2017

Michael - Thursday

I admit
that I stole the second half of the title from the article the appeared in The
Guardian today (that you can read HERE). It seemed too good to miss. The
article is about some recent work on an ancient Indian text called the
Bakhshali manuscript which is written in a form of Sanskrit on multiple pieces of tree bark. It seems
to be some sort of training manual for traders. It’s
quite impressive in its scope, including problems (posed in verse) in
arithmetic, algebra, and geometry. The solutions are given as well (and are
correct). And many of the numbers in the text contain zeroes, which are
indicated by heavy dots, showing placeholders in a number position that should
be 0. So what?

Section of the Bakhshali manuscript

Number
systems have been around for a very long time, of course. Most cultures
developed some way of representing numbers (if they wrote things down at all).
We’re all familiar with the Roman system that is used to this day, in a rather pretentious
fashion, for the year. It’s a horrible system, involving a few symbols and then
implied addition and subtraction from them—for example IX is one less than ten,
so nine, while XI is one more than ten so eleven. Some scholars have speculated
that the lack of a decent number system was a not insignificant problem for the
Roman Empire. The ancient Greeks—often put forward as the originators of modern
mathematics— had no concept of zero and had philosophical concerns about how
nothing could be something.

In fact,
the western world was in no way the leader in mathematics historically. That
honor belonged to other cultures and the Indian subcontinent has a strong
claim. Around 700 AD when an Indian astronomer had formalized the numeral zero,
and the concept was introduced to Europe, it was dismissed. Why have a number
for nothing? Why would you want to count nothing? It was like writing IIIIIV!
What was the point? The point was that it now allowed the sort of number system
we have today, where any number can be written in a way in which its value is
obvious in powers of ten (or two if you prefer that as computers do) and on which we
can do arithmetic easily. (Try the sum MMMXIV + MCLIX.)

Since this
is now so obvious to us, why was it ever hard? The idea of nothing, and the understanding
that if you have five sheep and sell five sheep then you have none left was always obvious. But why a symbol? It really isn’t that obvious at all. Why write down
that you have 0 sheep? You simply don’t have any! Children don’t find this
easy, by the way. Zero and negative
numbers are confusing things when you first meet them.

What has
happened recently is that the Bakhshali manuscript has been radiocarbon dated
with surprising results. Firstly, the text is not a part of one document at all,
but bits of different ones hundreds of years apart. Were they part of some sort
of ‘mathematics library’ that collected the material together? How did a part
of that collection end up buried near the village of Bakhshali in what is now
Pakistan? Dan Brown could make something of this, I’m sure. But the really
interesting part, at least to people interested in the history of mathematics, is
that one section of the document dates to around 300 AD. That’s about 500 years
earlier than the current oldest document using a zero symbol as a digit in a
decimal number representation. Looks pretty certain now that we owe essentially
everything we do these days with numbers to the people of the Indian subcontinent.

I knew there was something wrong with head, Michael. Thanks to you, I now know what. It's that, unlike most people, I like having nothing in it. As a young student, I loved negative numbers and got along quite well with them right from the start. Now, I will have to go off and figure out how such a friend of negative numbers as I turned out to be such an optimist. MMMM.